EDIT: this is directly from the Declaration of Independence:
> whenever any form of government becomes destructive of the ends for which it was established, it is the right of the people to alter or abolish it, and to institute new government
Note the any form of government.
I think this is exactly what would happen in the modern day.
But what is not going to happen is that a state gets to take it's ball and go home just because the nation had the sheer audacity to vote for the wrong President.
Which one are we talking about? Frankly, all the ones I've had the misfortune of living through, blur together for me. It's not like any of them have had any different policies. How could they, possibly?
If any area secedes, it might be because of "The President", but it won't be due to any particular one.
I.e. not mere "wah wah the government raised my taxes" but actually despotic, as Jefferson explains in the rest of the Declaration of Independence.
I'm pretty sure Lincoln would argue with you there.